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Reinsurance Optimal Design With Distortion Risk Measures And Risk Premiums
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Book Synopsis Reinsurance Optimal Design with Distortion Risk Measures and Risk Premiums by : Hirbod Assa
Download or read book Reinsurance Optimal Design with Distortion Risk Measures and Risk Premiums written by Hirbod Assa and published by . This book was released on 2015 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we consider the problem of optimal reinsurance design for general distortion risk measures and premiums. In the first part of the paper, we find the Lagrangian dual of the primal optimal reinsurance problem and show the strong duality holds. Therefore we characterize the optimal reinsurance policies by solving the dual problem and we will see that the solutions always have a multilayer structure. In addition we will see that for particular risk measures VaR and CVaR the optimal solutions are stop-loss policies. In the second part we focus our attention to reinsurance policies that are usually traded in the market, namely stop-loss, stop-loss after quota share and quota-share after stop-loss. We show how by one can find the optimal retentions by checking Karush-Kuhn-Tucker conditions. At the end, we study the particular cases VaR or CVaR.
Book Synopsis On Optimal Reinsurance Policy with Distortion Risk Measures and Premiums by : Hirbod Assa
Download or read book On Optimal Reinsurance Policy with Distortion Risk Measures and Premiums written by Hirbod Assa and published by . This book was released on 2014 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we consider the problem of optimal reinsurance design, when the risk is measured by a distortion risk measure and the premium is given by a distortion risk premium. First, we show how the optimal reinsurance design for the ceding company, the reinsurance company and the social planner can be formulated in the same way. Second, by introducing the “marginal indemnification functions”, we characterize the optimal reinsurance contracts. We show that, for an optimal policy, the associated marginal indemnification function only takes the values zero and one. We will see how the roles of the market preferences and premiums and that of the total risk are separated.
Book Synopsis Risk Measures and Optimal Reinsurance by : Fangda Liu
Download or read book Risk Measures and Optimal Reinsurance written by Fangda Liu and published by . This book was released on 2015 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we study the optimal reinsurance design problem and extend the classical model in three different directions: (1) In the first framework, we add the additional assumption that the reinsurer can default on its obligations. If the indemnity is beyond the reinsurer's payment ability, the reinsurer fails to pay for the exceeding part and this induces a default risk for the insurer. In our model, the reinsurer is assumed to measure the risk of an insured loss by Value-at-Risk regulation and prepares the same amount of money as the initial reserve. As soon as the indemnity is larger than this value plus the premium, default occurs. From the insurer's point of view, two optimization problems are going to be considered when the insurer: 1) maximizes his expectation of utility; 2) minimizes the VaR of his retained loss. (2) In the second framework, the reinsurance buyer (insurer) adopts a convex risk measure to control his total loss while the reinsurance seller (reinsurer) price the reinsurance contract by Wang's premium principle with a distortion. Without specifying a particular convex risk measure and distortion, we obtain a general expression for the optimal reinsurance contract that minimizes the insurer's total risk exposure. (3) In the third framework, we study optimal reinsurance designs from the perspectives of both an insurer and a reinsurer and take into account both an insurer's aims and a reinsurer's goals in reinsurance contract designs. We develop optimal reinsurance contracts that minimize the convex combination of the VaR risk measures of the insurer's loss and the reinsurer's loss under two types of constraints, respectively. The constraints describe the interest of both the insurer and the reinsurer. With the first type of constraints, the insurer and the reinsurer each have their limit on the VaR of their own loss. With the second type of constraints, the insurer has a limit on the VaR of his loss while the reinsurer has a target on his profit from selling a reinsurance contract. For both types of constraints, we derive the optimal reinsurance form for a wide class of reinsurance policies and under the expected value reinsurance premium principle.
Book Synopsis Optimal Reinsurance with One Insurer and Multiple Reinsurers by : Tim J. Boonen
Download or read book Optimal Reinsurance with One Insurer and Multiple Reinsurers written by Tim J. Boonen and published by . This book was released on 2015 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we consider a one-period optimal reinsurance design model with n reinsurers and an insurer. For very general preferences of the insurer, we obtain that there exists a very intuitive pricing formula for all reinsurers that use a distortion premium principle. The insurer determines its optimal risk that it wants to reinsure via this pricing formula. This risk it wants to reinsure is then shared by the reinsurers via tranching. The optimal ceded loss functions among multiple reinsurers are derived explicitly under the additional assumptions that the insurer's preferences are given by an inverse-S shaped distortion risk measure and that the reinsurer's premium principles are some functions of the Conditional Value-at-Risk. We also demonstrate that under some prescribed conditions, it is never optimal for the insurer to cede its risk to more than two reinsurers.
Book Synopsis On Pareto-Optimal Reinsurance with Constraints Under Distortion Risk Measures by : Wenjun Jiang
Download or read book On Pareto-Optimal Reinsurance with Constraints Under Distortion Risk Measures written by Wenjun Jiang and published by . This book was released on 2017 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper studies the Pareto-optimal reinsurance policies, where both the insurer's and the reinsurer's risks and returns are considered. We assume that the risks of the insurer and the reinsurer, as well as the reinsurance premium, are determined by some distortion risk measures with different distortion operators. Under the constraint that a reinsurance policy is feasible only if the resulting risk of each party is below some pre-determined values, we derive explicit expressions for the optimal reinsurance polices. Methodologically, we show that the generalized Neyman-Pearson method, the Lagrange multiplier method, and the dynamic control methods can be utilized to solve the optimization problem with constraints. Special cases when both parties' risks are measured by Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) are studied in great details. Numerical examples are provided to illustrate practical implications of the results.
Book Synopsis Topics in Optimal Reinsurance Design, Risk Measures, and Forward Performance Processes by : Wing Fung Chong
Download or read book Topics in Optimal Reinsurance Design, Risk Measures, and Forward Performance Processes written by Wing Fung Chong and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Topics in Optimal Reinsurance Design, Risk Measures, and Forward Performance Processes by : 莊榮峰
Download or read book Topics in Optimal Reinsurance Design, Risk Measures, and Forward Performance Processes written by 莊榮峰 and published by . This book was released on 2017 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Optimal Reinsurance Designs by : Chengguo Weng
Download or read book Optimal Reinsurance Designs written by Chengguo Weng and published by . This book was released on 2009 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: The research on optimal reinsurance design dated back to the 1960's. For nearly half a century, the quest for optimal reinsurance designs has remained a fascinating subject, drawing significant interests from both academicians and practitioners. Its fascination lies in its potential as an effective risk management tool for the insurers. There are many ways of formulating the optimal design of reinsurance, depending on the chosen objective and constraints. In this thesis, we address the problem of optimal reinsurance designs from an insurer's perspective. For an insurer, an appropriate use of the reinsurance helps to reduce the adverse risk exposure and improve the overall viability of the underlying business. On the other hand, reinsurance incurs additional cost to the insurer in the form of reinsurance premium. This implies a classical risk and reward tradeoff faced by the insurer. The primary objective of the thesis is to develop theoretically sound and yet practical solution in the quest for optimal reinsurance designs. In order to achieve such an objective, this thesis is divided into two parts. In the first part, a number of reinsurance models are developed and their optimal reinsurance treaties are derived explicitly. This part focuses on the risk measure minimization reinsurance models and discusses the optimal reinsurance treaties by exploiting two of the most common risk measures known as the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE). Some additional important economic factors such as the reinsurance premium budget, the insurer's profitability are also considered. The second part proposes an innovative method in formulating the reinsurance models, which we refer as the empirical approach since it exploits explicitly the insurer's empirical loss data. The empirical approach has the advantage that it is practical and intuitively appealing. This approach is motivated by the difficulty that the reinsurance models are often infinite dimensional optimization problems and hence the explicit solutions are achievable only in some special cases. The empirical approach effectively reformulates the optimal reinsurance problem into a finite dimensional optimization problem. Furthermore, we demonstrate that the second-order conic programming can be used to obtain the optimal solutions for a wide range of reinsurance models formulated by the empirical approach.
Book Synopsis Budget-Constrained Optimal Reinsurance Design Under Coherent Risk Measures by : Ka Chun Cheung
Download or read book Budget-Constrained Optimal Reinsurance Design Under Coherent Risk Measures written by Ka Chun Cheung and published by . This book was released on 2017 with total page 25 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reinsurance is a versatile risk management strategy commonly employed by insurers to optimize their risk profile. In this paper, we study an optimal reinsurance design problem minimizing a general law-invariant coherent risk measure of the net risk exposure of a generic insurer, in conjunction with a general law-invariant comonotonic additive convex reinsurance premium principle and a premium budget constraint. Due to its intrinsic generality, this contract design problem encompasses a wide body of optimal reinsurance models encountered in practice. A three-step solution scheme is presented. Firstly, the objective and constraint functions are exhibited in the so-called Kusuoka's integral representations. Secondly, the mini-max theorem for infinite dimensional spaces is applied to interchange the infimum on the space of indemnities and the supremum on the space of probability measures. Thirdly, the recently developed Neyman-Pearson methodology due to Lo (2017) is adopted to solve the resulting infimum problem. Analytic and transparent expressions for the optimal reinsurance policy are provided, followed by illustrative examples.
Book Synopsis Optimal Reinsurance in a Market of Multiple Reinsurers Under Law-Invariant Convex Risk Measures by : Jun Cai
Download or read book Optimal Reinsurance in a Market of Multiple Reinsurers Under Law-Invariant Convex Risk Measures written by Jun Cai and published by . This book was released on 2017 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is natural to connect reinsurance problems with risk measures since a reinsurance contract is an efficient risk management tool for an insurer and the reinsurance premium can also be viewed as a measure of a reinsurer's risk. In this paper, we assume that the insurer uses a law-invariant convex risk measure, while reinsurers use a Wang's premium principle to determine their premiums. We study an optimal reinsurance policy design from an insurer's perspective in a market of multiple reinsurers. Both the insurer's risk measure and the reinsurer's premium principle represent broad families of risk measures with considerable generality. We provide a general formula for the optimal solution which recovers existing results if particular law-invariant convex measures, such as the AVaR, and particular premium principles are assigned.
Book Synopsis Optimal Reinsurance by : Ka-Chun Joseph Sung
Download or read book Optimal Reinsurance written by Ka-Chun Joseph Sung and published by Open Dissertation Press. This book was released on 2017-01-26 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation, "Optimal Reinsurance: a Contemporary Perspective" by Ka-chun, Joseph, Sung, 宋家俊, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: In recent years, general risk measures have played an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance problems using risk measures as yard sticks beyond the classical expected utility framework. In this thesis, the stop-loss reinsurance is first shown to be an optimal contract under law-invariant convex risk measures via a new simple geometric argument. This similar approach is then used to tackle the same optimal reinsurance problem under Value at Risk and Conditional Tail Expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution in these cases. These two results hint that law-invariant convex risk measure may be better and more robust to expected larger claims than Value at Risk and Conditional Tail Expectation even though they are more commonly used. In addition, the problem of optimal reinsurance design for a basket of n insurable risks is studied. Without assuming any particular dependence structure, a minimax optimal reinsurance decision formulation for the problem has been successfully proposed. To solve it, the least favorable dependence structure is first identified, and then the stop-loss reinsurances are shown to minimize a general law-invariant convex risk measure of the total retained risk. Sufficient condition for ordering the optimal deductibles are also obtained. Next, a Principal-Agent model is adopted to describe a monopolistic reinsurance market with adverse selection. Under the asymmetry of information, the reinsurer (the principal) aims to maximize the average profit by selling a tailor-made reinsurance to every insurer (agent) from a (huge) family with hidden characteristics. In regard to Basel Capital Accord, each insurer uses Value at Risk as the risk assessment, and also takes the right to choose different risk tolerances. By utilizing the special features of insurance layers, their optimality as the first-best strategy over all feasible reinsurances is proved. Also, the same optimal reinsurance screening problem is studied under other subclass of reinsurances: (i) deductible contracts; (ii) quota-share reinsurances; and (iii) reinsurance contracts with convex indemnity, with the aid of indirect utility functions. In particular, the optimal indirect utility function is shown to be of the stop-loss form under both classes (i) and (ii); while on the other hand, its non-stop-loss nature under class (iii) is revealed. Lastly, a class of nonzero-sum stochastic differential reinsurance games between two insurance companies is studied. Each insurance company is assumed to maximize the difference of the opponent's terminal surplus from that of its own by properly arranging its reinsurance schedule. The surplus process of each insurance company is modeled by a mixed regime-switching Cramer-Lundberg approximation. It is a diffusion risk process with coefficients being modulated by both a continuous-time finite-state Markov Chain and another diffusion process; and correlations among these surplus processes are allowed. In contrast to the tradit
Book Synopsis Optimal Reinsurance with Multiple Reinsurers by : Tim J. Boonen
Download or read book Optimal Reinsurance with Multiple Reinsurers written by Tim J. Boonen and published by . This book was released on 2018 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study economic pricing of reinsurance contracts via competition of an insurer with multiple reinsurers. All firms are endowed with distortion risk measures or expected exponential utilities. We require that contracts are Pareto optimal, individually rational, and satisfy a competition constraint that we call coalition stability. Indemnities are characterized by imposing Pareto optimality, as studied in the literature. In this paper, we characterize the corresponding premiums. There is a gain for the insurer due to the competition constraint. When the firms use distortion risk measures, this constraint yields stability for subcoalitions, which is a condition akin to the core in cooperative game theory. We show this gain for the insurer in closed form. Then, we derive that the premium is represented by a distortion premium function. If the firms use expected exponential utilities, the premium is represented by an exponential premium. We illustrate this premium function with the Mean Conditional Value-at-Risk.
Book Synopsis Systemic Risk and Reinsurance by : Weidong Tian
Download or read book Systemic Risk and Reinsurance written by Weidong Tian and published by MDPI. This book was released on 2020-07-01 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Special Issue covers the topic of timely vital risk management - systemic risk - from many important perspectives. It includes novel and scientific approaches from the network with topological indicators on systemic risk, community analysis of the global financial system, welfare analysis of capital insurance and the impact of capital requirement, risk measures, and optimal portfolio and optimal reinsurance under risk constraint. Most articles study the financial sector and insurance companies after the financial crisis of 2008–2009 circa ten years prior. The COVID-19 global pandemic in 2020 has caused similar or even greater challenges for the entire economy. Therefore, this Special Issue will be useful for anyone interested in systemic risk management.
Book Synopsis Reinsurance for Catastrophes and Cataclysms by : David M. Cutler
Download or read book Reinsurance for Catastrophes and Cataclysms written by David M. Cutler and published by . This book was released on 1997 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper examines the optimal design of insurance and reinsurance policies. We first consider reinsurance for catastrophes: risks which are large for any one insurer but not for the reinsurance market as a whole. Reinsurance for catastrophes is complicated by adverse selection. Optimal reinsurnace in the presence of adverse selection depends critically on the source of information asymmetry. When information on the probability of a loss is private but the magnitude of the loss is public optimal reinsurance employs a deductible-style deductible-style excess-of-loss policy, and when is is private but the proba- bility of a loss is common, optimal reinsurance covers small and large risks, but makes the primary insurer responsible for moderate risks. There is a dramatic divergence between these designs, which suggests that traditional approaches to design may be misguided. We then consider reinsurance for cata- clysms: risks that are so large that a loss can threaten the solvency of re- insurance such as a major earthquake, while others derive from common risks-changes in conditions that affect many individuals-such as the liability revolution or or escalating medical care costs. We argue that cataclysms must be reinsured in either broad securities markets or by the government. Beyond their one- period loss potential, cataclysms pose another risk: risk levels change over time. A simulation model traces the implications of evolving risk levels for long-term patterns of losses and premiums, where the latter reflect learning learning about loss distributions. Premium risk emerges as an important part of risk, which reinsurance and primary insurance markets do not adequately diversify."
Book Synopsis Marginal Indemnification Function Formulation for Optimal Reinsurance by : Sheng Chao Zhuang
Download or read book Marginal Indemnification Function Formulation for Optimal Reinsurance written by Sheng Chao Zhuang and published by . This book was released on 2017 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we propose to combine the Marginal Indemnification Function (MIF) formulation and the Lagrangian dual method to solve optimal reinsurance model with distortion risk measure and distortion reinsurance premium principle. The MIF method exploits the absolute continuity of admissible indemnification functions and formulates optimal reinsurance model into a functional linear programming of determining an optimal measurable function valued over a bounded interval. The MIF method was recently introduced to analyze the reinsurance model but without premium budget constraint. In this paper, a Lagrangian dual method is applied to combine with MIF to solve for optimal reinsurance solutions under premium budget constraint. Compared with the existing literature, the proposed integrated MIF-based Lagrangian dual method provides a more technically convenient and transparent solution to the optimal reinsurance design. To demonstrate the practicality of the proposed method, analytical solution is derived on a particular reinsurance model that involves minimizing Conditional Value at Risk (a special case of distortion function) and with the reinsurance premium being determined by the inverse-S shaped distortion principle.
Download or read book Reinsurance written by Hansjörg Albrecher and published by John Wiley & Sons. This book was released on 2017-08-21 with total page 501 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reinsurance: Actuarial and Statistical Aspects provides a survey of both the academic literature in the field as well as challenges appearing in reinsurance practice and puts the two in perspective. The book is written for researchers with an interest in reinsurance problems, for graduate students with a basic knowledge of probability and statistics as well as for reinsurance practitioners. The focus of the book is on modelling together with the statistical challenges that go along with it. The discussed statistical approaches are illustrated alongside six case studies of insurance loss data sets, ranging from MTPL over fire to storm and flood loss data. Some of the presented material also contains new results that have not yet been published in the research literature. An extensive bibliography provides readers with links for further study.
Book Synopsis Pareto-optimal Reinsurance Policies with Maximal Synergy by : Wenjun Jiang
Download or read book Pareto-optimal Reinsurance Policies with Maximal Synergy written by Wenjun Jiang and published by . This book was released on 2019 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimal reinsurance policies have been studied extensively in the economics and insurance literature. Two types of optimality criteria are commonly used: maximizing the expected utility (EU) or minimizing risks. Understandably, applying the two types of criteria usually will result in different “optimal” policies. To strike a balance between maximizing utility and minimizing risk, Borch (1960b) derived the EU-maximizing reinsurance policies assuming that the admissible policies are those that minimize the total variance of the losses borne by the two parties. This in fact implies that only quota-share policies are admissible and greatly simplifies the problem. In this paper, we follow the approach in Borch (1960b). However, we assume that the two parties apply distortion risk measures instead of variance. We first identify a set of reinsurance policies that minimize the total risk shared by the two parties, then we take this set of policies as admissible and determine the Pareto-optimal policies that maximize the EU of the two parties. In contrast to the results in Borch (1960b), we show that applying risk measures such as the Value at Risk (VaR) and the Tail Value at Risk (TVaR) results in multi-layered policies.
